138 EOOK OF MATHEMATICAL PROBLEMS. 



It may be noticed also that we obtain the following equation for //. 

 kf 



equivalent to U- - a\( - b\ = c /2 ; 



whose roots are the squares of the semi axes. 



To each root correspond two foci, which are real for one and 

 unreal for the other. 



The same method will apply to all cases ; and, the foci being 

 found, the directrices are their polars. 



The more useful forms of the equation are 



which, for different values of A, represents a series of conies 

 passing through four given points; two of the joining lines 

 being taken as axes : 



x y 



the equation of a conic touching the axes at distances h, k re- 

 spectively from the origin. It is sometimes convenient to use 

 this as the equation of the conic touching four given straight 

 lines ; h, k, A being then parameters connected by the equations 

 1 1\/1 1\ A 2 /I 1\/1 1 



the equations of the other two given straight lines being 



x + y -l x 



"T^-=l. -. 



a b a! 



When (2) represents a parabola, A = 1 ; and the equation may 

 be written 



The equation of the polar of (X t Y) to the conic in the most 

 general form is 



